Can all subsequences of a function sequence in C([0,1], R) converge uniformly?

[Malmstrom]

Let $$y_n$$ be a sequence of functions in $$\mathcal{C}([0,1], \mathbb{R})$$
Suppose that every subsequence of $$y_n$$ has a subsequence that converges uniformly. Prove that they all converge to the same limit.

 

[Landau]

So, what have you tried?

 

[Malmstrom]

This is not an exercise. This is a question that rose solving an exercise. Can't solve it.